Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [exponential-distribution]

To be used for questions on using, finding, or otherwise relating to Exponential Distributions.

For an Exponential distribution as a probability density function:

$f(x;\lambda) =\lambda e^{-\lambda x}\quad$ for $x \ge 0 $

and

$f(x;\lambda) =0\quad$ for $x \lt 0 $

where $\lambda$ is the rate parameter.

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Pdf of the difference of two exponentially distributed random variables

Suppose we have two independent random variables $Y$ and $X$, both being exponentially distributed with respective parameters $\mu$ and $\lambda$. How can we calculate the pdf of $Y-X$?
Mathematics Lover
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How to prove that minimum of two exponential random variables is another exponential random variable?

How can I prove that the minimum of two exponential random variables is another exponential random variable, i.e. Z = min(X,Y)
user82004
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How to prove that geometric distributions converge to an exponential distribution?

How to prove that geometric distributions converge to an exponential distribution? To solve this, I am trying to define an indexing $n$/$m$ and to send $m$ to infinity, but I get zero, not some relevant distribution. What is the technique or…
Ramamurthy Shankar
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Gamma Distribution out of sum of exponential random variables

I have a sequence $T_1,T_2,\ldots$ of independent exponential random variables with paramter $\lambda$. I take the sum $S=\sum_{i=1}^n T_i$ and now I would like to calculate the probability density function. Well, I know that $P(T_i>t)=e^{-\lambda…
TI Jones
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What's the kurtosis of exponential distribution?

Original question (with confused terms): Wikipedia and Wolfram Math World claim that the kurtosis of exponential distribution is equal to $6$. Whenever I calculate the kurtosis in math software (or manually) I get $9$, so I am slightly confused. I…
Szymon Brych
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On the proof that every positive continuous random variable with the memoryless property is exponentially distributed

The theorem to prove is: $X$ is a positive continuous random variable with the memoryless property, then $X \sim Expo(\lambda)$ for some $\lambda$. The proof is explained in this video, but I will type it out here as well. I would like to get some…
jlcv
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What is the difference between a Poisson and an Exponential distribution?

For a Poisson distribution: $$\mathsf{P}(X=x)=\frac{e^{-\mu}\times \mu^x}{x!}$$ where $\mu$ is the mean number of occurrences. For an Exponential distribution: $$f(x;\lambda) = \begin{cases} \lambda e^{-\lambda x} & x \ge 0 \\ 0 & x <…
BLAZE
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The time it takes for a candle having a lifetime that follows an exponential distribution to go off

We have $5$ candles each having a lifetime which follows an exponential distribution with parameter $\lambda$. We light up each candle at time $t=0$. Assume that $Y$ is the time that it takes for the third candle to go off. What is the expectation…
Jared Garreta
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The expectation of $e^X \left(1-(1-e^{-X}\right)^n)$ when $X$ has Exponential Distribution

To my surprise, I was able to evaluate the following expression in Mathematica: $$E\left[e^X \left(1-(1-e^{-X}\right)^n) \right] = \frac{y}{y-1} \left(1-\frac{1}{\binom{n+y-1}{y-1}}\right)\quad X\sim\text{Exp}(y)$$ with the right hand side being…
Thomas Ahle
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First moments of Geometric Brownian Motion-like process with non-normal shocks

First consider a standard GBM process of the form, $$\frac{dS_t}{S_t} = \mu \, dt+ \sigma \, dW_t$$ but instead of the normal $W_t \sim N(0,1)$ , instead we have that, $$W_t \sim \operatorname{EMG}^-(0,1,\lambda)$$ where $\operatorname{EMG}^-$ is…
sfortney
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Distribution of sum of exponential variables with different parameters

We have $k$ independent random variables with exponential distribution ($T_1, T_2, \ldots , T_k$), parameters of random variables are ($\lambda,\frac{\lambda}{2},\frac{\lambda}{3},\ldots,\frac{\lambda}{k}$), what is the distribution of new variable…
user137927
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Finding UMVUE of $\theta$ when the underlying distribution is exponential distribution

Hi I'm solving some exercise problems in my text : "A Course in Mathematical Statistics". I'm in the chapter "Point estimation" now, and I want to find a UMVUE of $\theta$ where $X_1 ,...,X_n$ are i.i.d random variables with the p.d.f $f(x;…
user88914
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Conditional expectation involving some complications around exponential random variables

Here is my problem. Consider four independent exponential distributions $X^A_1$, $X^B_1$, $X^A_2$, $X^B_2$ where $X^A_1$ and $X^B_1$ are $\exp(\lambda_1)$ and $X^A_2$ and $X^B_2$ are $\exp(\lambda_2)$. There is another random variable $\mu$ where…
ykdec
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Prove that if $X\sim\mathcal{N}(0,1)$, then $X^2-1$ is subexponential

Definition: A random variable $X$ with $E[X]=0$ is called sub-exponential with parameters $(\nu,\alpha)$ iff for each $\lambda$ satisfying $|\lambda|<\frac{1}{\alpha}$, we have: $$E[e^{\lambda X}]\le e^{\frac{\lambda^2 \nu^2}{2}} $$ We write this as…
Arman Malekzadeh
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Let $U=\operatorname{min}\{X,Y\}$ and $V=\operatorname{max}\{X,Y\}$. Show that $V-U$ is independent of $U$.

Let $X$ and $Y$ be exponentially distributed random variables with parameter $1$ and let $U=\operatorname{min}\{X,Y\}$ and $V=\operatorname{max}\{X,Y\}$. Show that $V-U$ is independent of $U$. We have shown that $U$ is distributed…
Aka_aka_aka_ak
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